Mathematicians have long been fascinated by the concept of aperiodic tiling: covering a whole surface in shapes without creating a repeating design. On these surfaces, there is no subsection that could be copied and pasted to expand the plane further. Past successes in aperiodic tiling have always featured a minimum of two differently shaped tiles to create the non-repeating patterns, but the search for a single ‘einstein’ shape - not referring to the physicist Albert Einstein, but from the German ‘one stone’ – that can uniquely cover an infinite surface on its own, continued.

Above. Repeating tiles of “the hat,” a 13-sided polygon that was confirmed in 2023 to be the first einstein tile.

Below. A few months after the discovery of “the hat,” a “vampire einstein” shape was proposed which can aperiodically cover an infinite surface even without its mirror image.

The first einstein shape was identified in 2023 by systems engineer David Smith and was nicknamed the “hat.” The “hat” is a 13-sided polygon made up of smaller kite-like shapes merged together. By combining an infinite number of “hats” and their mirror images, an endless non-repeating array can be formed – but this had to be confirmed by several rounds of simulations involving computer scientist Craig Kaplan, software developer Joseph Samuel Myers, and mathematician Chaim Goodman-Strauss.  

Although the “hat” was a once-in-a-lifetime discovery in itself representing the culmination of decades of study*, it was soon joined by another einstein tile only a few months later. The same team uncovered a new shape that could be lain out in a non-repeating pattern even without the use of its mirror image. In the absence of the shape’s reflection, it was nicknamed the “vampire” einstein. The researchers were stunned to find that the first identified einsteins could take such simple forms. The “vampire” einstein is based on a 14-sided polygon with modified, curved edges. The discovery of the shape opened up a whole family of other “vampire” monotiles known as “Spectres.”

*In 1966, mathematician Robert Berger identified a set of 104 unique tiles that could be combined to create an aperiodic mosaic. Five years later, physicist Roger Penrose showed that the aperiodic condition could still be met using only 2 unique tiles.

What does this mean for non-mathematicians? Knowing that simple shapes could be aperiodic (non-repeating) could have major implications for materials science, a field of engineering that identifies materials used for construction and manufacturing. Chemist Daniel Schectman was awarded the Nobel Prize for his 1982 discovery of quasicrystals: crystals whose atoms are packed in a non-repeating pattern. The non-repeating nature of quasicrystals makes them extremely strong but also poor conductors of heat. They have since been used in blends of steel for making surgical instruments and are being considered as insulators in engines. How einstein shapes affect material science remains to be seen, but these aperiodic discoveries represent an exciting new frontier across different fields of study.